Model-agnostic meta-learning (MAML) has emerged as one of the most successful meta-learning techniques in few-shot learning. It enables us to learn a $\textit{meta-initialization}$ of model parameters (that we call $\textit{meta-model}$) to rapidly adapt to new tasks using a small amount of labeled training data. Despite the generalization power of the meta-model, it remains elusive that how $\textit{adversarial robustness}$ can be maintained by MAML in few-shot learning. In addition to generalization, robustness is also desired for a meta-model to defend adversarial examples (attacks). Toward promoting adversarial robustness in MAML, we first study $\textit{when}$ a robustness-promoting regularization should be incorporated, given the fact that MAML adopts a bi-level (fine-tuning vs. meta-update) learning procedure. We show that robustifying the meta-update stage is sufficient to make robustness adapted to the task-specific fine-tuning stage even if the latter uses a standard training protocol. We also make additional justification on the acquired robustness adaptation by peering into the interpretability of neurons' activation maps. Furthermore, we investigate $\textit{how}$ robust regularization can $\textit{efficiently}$ be designed in MAML. We propose a general but easily-optimized robustness-regularized meta-learning framework, which allows the use of unlabeled data augmentation, fast adversarial attack generation, and computationally-light fine-tuning. In particular, we for the first time show that the auxiliary contrastive learning task can enhance the adversarial robustness of MAML. Finally, extensive experiments are conducted to demonstrate the effectiveness of our proposed methods in robust few-shot learning.